(New page: ==Signal== Let the signal be <math>x(t) = 5cos(3\pi t) + sin(\pi t)\,</math> ==Analysis== First rewrite the signal as a sum of complex exponentials: :<math> x(t) = 5(\frac{e^{3\pi jt} + e^...) |
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==Signal== | ==Signal== | ||
Let the signal be <math>x(t) = 5cos(3\pi t) + sin(\pi t)\,</math> | Let the signal be <math>x(t) = 5cos(3\pi t) + sin(\pi t)\,</math> | ||
| + | Find the Fourier Series coefficients | ||
==Analysis== | ==Analysis== | ||
First rewrite the signal as a sum of complex exponentials: | First rewrite the signal as a sum of complex exponentials: | ||
Revision as of 13:24, 26 September 2008
Signal
Let the signal be $ x(t) = 5cos(3\pi t) + sin(\pi t)\, $ Find the Fourier Series coefficients
Analysis
First rewrite the signal as a sum of complex exponentials:
- $ x(t) = 5(\frac{e^{3\pi jt} + e^{-3\pi jt}}{2}) + \frac{e^{\pi jt} - e^{-\pi jt}}{2j} $
Simplifying gives:
- $ x(t) = \frac{5}{2} e^{3\pi jt} + \frac{5}{2} e^{-3\pi jt} + \frac{1}{2j} e^{\pi jt} - \frac{1}{2j} e^{-\pi jt} $
The fundamental period is:
- $ \omega_o = \frac{2\pi}{T} = \pi $, with $ T = 2\, $ being the period of the original signal.
From the fundamental period, it is easily seen that the fourier series coefficients are:
- $ a_{-3} = \frac{5}{2} $
- $ a_{-1} = -\frac{1}{2j} $
- $ a_{1} = \frac{1}{2j} $
- $ a_{3} = \frac{5}{2} $
- $ a_{k} = 0\, $, for all other k
